Theory Building and Causal Processes in Poitical Science

PSCI 3300.003 Political Science Research Methods

A. Jordan Nafa

University of North Texas

September 27th, 2022

Overview

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Review of Key Terms

  • Potential Outcomes: A way of formally expressing each of the possible outcomes unit \(i\) could experience under a given treatment regime

  • Causal Effect: The change we would expect to observe if we altered some feature of the world compared to what would have happened in the absence of such a manipulation

  • Estimand: The causal effect we are trying to estimate. The estimand connect research question, theory, and statistics.

  • Directed Acyclic Graphs (DAGs): Non-parametric graphical representations of causal relationships.

Types of Data

  • Experimental

    • You have direct control over which units get treatment and can thus rely on random assignment to identify the causal path \(\mathrm{\treat{X}_{\obs{i}}} \longrightarrow \mathrm{\resp{Y}_{\obs{i}}}\)

  • Observational

    • You don’t have control over which units get assigned the treatment and thus cannot rely on random assignment to identify causal effects

    • This makes causal inference difficult, but by no means impossible, in many areas of political science and most of international relations

  • Common misconception that experiments can “prove” causality

    • For both practical and epistemological reasons, it is impossible to “prove” a causal relationship and you should avoid using the word

Causal Diagrams

  • Directed Acyclic Graphs (DAGs)

    • Directed

      • Each node has an arrow that points to another node

    • Acyclic

      • You can’t cycle back to a node and arrows only have one direction

    • Graph

      • It’s… um… a graph

Causal Diagrams

  • DAGs are a type of non-parametric graphical causal model

    • Graphical model of the data generation process (DGP)

    • Maps your theory of the process to a model

    • Fancy math called “do-calculus” tells you what to adjust for to isolate and identify causation

      • More importantly, perhaps, it tells you what not to adjust for

    • Makes our assumptions about the causal process explicit

Acyclicity

  • What if there’s something that really is cyclical?

    • \(\mathrm{Democracy} \longrightarrow \mathrm{Development} \longrightarrow \mathrm{Democracy}\)

    • This isn’t acyclic!

    • \(\mathrm{Democracy} \longleftrightarrow \mathrm{Development}\)

    • Remember the causal ordering assumption we talked about last time

      • No reverse causality or simultaneity

    • If a process is truly cyclical, we can represent that using a time index

      • \(\mathrm{Democracy_{\tim{t-1}}} \longrightarrow \mathrm{Development_{\tim{t}}} \longrightarrow \mathrm{Democracy_{\tim{t+1}}}\)

Causal Graphs and Theory Building

List Variables

Imagine we are interested in whether the adoption of gender quotas in has a causal effect on the level of female representation in a country’s government.

  • Our first step is to make a list of the theoretically relevant variables.

  • Gender quota is the treatment and female representation is the outcome of interest

    • \(\mathrm{\treat{Quota}_{\tim{t}}} \longrightarrow \mathrm{\resp{Female~Representation}_{\tim{t}}}\) at each time \(\tim{t} \in \{1,2,\dots,\tim{T}\}\)

  • What other factors could influence Quota adoption, female representation in government, or both?

List Variables

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

  • Political Corruption \(\covar{Z}_{\tim{t}}\) has a causal effect on Female Representation and Quota Adoption

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

  • Political Corruption \(\covar{Z}_{\tim{t}}\) has a causal effect on Female Representation and Quota Adoption

  • Electoral institutions \(\covar{V}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

  • Political Corruption \(\covar{Z}_{\tim{t}}\) has a causal effect on Female Representation and Quota Adoption

  • Electoral institutions \(\covar{V}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Women’s civil society organizations \(\covar{W}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

  • Political Corruption \(\covar{Z}_{\tim{t}}\) has a causal effect on Female Representation and Quota Adoption

  • Electoral institutions \(\covar{V}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Women’s civil society organizations \(\covar{W}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Attitudes towards women in politics \(\covar{U}_{\tim{t}}\) are affected by Quota Adoption and have an effect on Female Representation

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

  • Political Corruption \(\covar{Z}_{\tim{t}}\) has a causal effect on Female Representation and Quota Adoption

  • Electoral institutions \(\covar{V}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Women’s civil society organizations \(\covar{W}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Attitudes towards women in politics \(\covar{U}_{\tim{t}}\) are affected by Quota Adoption and have an effect on Female Representation

  • Women’s civil society organizations \(\covar{W}_{\tim{t}}\) have an effect on attitudes towards women in politics \(\covar{U}_{\tim{t}}\)

Make Connections

  • Gender Quotas \(\treat{X}_{\tim{t}}\) cause an increase in Female Representation \(\resp{Y}_{\tim{t}}\)

  • Political Corruption \(\covar{Z}_{\tim{t}}\) has a causal effect on Female Representation and Quota Adoption

  • Electoral institutions \(\covar{V}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Women’s civil society organizations \(\covar{W}_{\tim{t}}\) have an effect on Female Representation and Quota Adoption

  • Attitudes towards women in politics \(\covar{U}_{\tim{t}}\) are affected by Quota Adoption and have an effect on Female Representation

  • Women’s civil society organizations \(\covar{W}_{\tim{t}}\) have an effect on attitudes towards women in politics \(\covar{U}_{\tim{t}}\)

  • Time invariant differences \(\upsilon\)?

Make Connections

  • We can then express each node as function of those that influence it

    • \(\mathrm{\resp{Female~Representation}}_{\tim{t}} = f(\treat{X}_{\tim{t}}, \covar{Z}_{\tim{t}}, \covar{V}_{\tim{t}}, \covar{W}_{\tim{t}}, \covar{U}_{\tim{t}}, \upsilon)\)

    • \(\mathrm{\treat{Gender~Quota}}_{\tim{t}} = f(\covar{V}_{\tim{t}}, \covar{W}_{\tim{t}}, \covar{U}_{\tim{t}}, \covar{Z}_{\tim{t}})\)

    • \(\mathrm{\covar{Political~Corruption}}_{\tim{t}} = f(\covar{V}_{\tim{t}}, \upsilon, \dots)\)

    • \(\mathrm{\covar{Electoral~Institutions}}_{\tim{t}} = f(\upsilon, \dots)\)

    • \(\mathrm{\covar{Womens~CSOs}}_{\tim{t}} = f(\upsilon, \dots)\)

    • \(\mathrm{\covar{Gender~Attitudes}}_{\tim{t}} = f(\treat{X}_{\tim{t}}, \covar{W}_{\tim{t}}, \upsilon, \dots)\)

  • Which of these do we need to measure to identify the path \(\mathrm{\treat{Gender~Quota}}_{\tim{t}} \longrightarrow \mathrm{\resp{Female~Representation}}_{\tim{t}}\)?

Causal Identification

  • Using theory and existing research, we’ve identified possible relationships between concepts (nodes)

  • We only care about \(\treat{X}_{\tim{t}} \longrightarrow \resp{Y}_{\tim{t}}\), but what do we do about all the other nodes?

  • A causal effect is identified if the association between treatment and outcome is properly stripped and isolated

Paths and Associations

  • Arrows in a DAG transmit associations between nodes

  • We need to redirect and control these paths by adjusting or conditioning in order to isolate the relationship we are interested in

  • Without additional often strong assumptions, it is only possible to adjust for things we can observe

Types of Associations

Confounding

  • \(\treat{X}\) causes \(\resp{Y}\)

  • But \(\covar{Z}\) causes both \(\treat{X}\) and \(\resp{Y}\)

  • \(\covar{Z}\) confounds the \(\treat{X} \longrightarrow \resp{Y}\) association

  • Failing to adjust for the confounder \(\covar{Z}\) results in omitted variable bias

  • The causal path \(\treat{X} \longleftarrow \covar{Z} \longrightarrow \resp{Y}\) is called a backdoor

Closing Backdoors

  • \(\treat{X}\) and \(\resp{Y}\) are d-connected because associations can pass through a third variable, \(\covar{Z}\)

  • The relationship between \(\treat{X}\) and \(\resp{Y}\) is not causally identified unless we close the backdoor created by \(\covar{Z}\)

  • To do this, we need to somehow adjust for \(\covar{Z}\)

  • Most common method of adjustment is to include \(\covar{Z}\) in a regression model

Closing Backdoors

  • Returning to our earlier example, what do we need to adjust for to identify the causal path \(\treat{X}_{\tim{t}} \longrightarrow \resp{Y}_{\tim{t}}\) under the assumed data generation process?

  • Answer is fairly straightforward in this example, but for illustrative purposes we’ll use the dagitty and ggdag packages to figure it out

Closing Backdoors

Our first step is to define the DAG for our theoretical model

# Define the DAG for the causal relationship
quota_dag <- dagify(
  Y ~ X + Z + V + W + U + C,
  Z ~ V + C,
  V ~ C,
  X ~ V + W + Z,
  U ~ X + W + C,
  W ~ C,
  labels = list(X = "X[t]", Y = "Y[t]", Z = "Z[t]", V = "V[t]", 
                W = "W[t]", U = "U[t]", C = "upsilon"),
  coords = list(
    x = c(X = 0, Y = 1, Z = 1, V = 0, W  = 0, U = 1, C = 0.5),
    y = c(X = 0, Y = 0, Z = 0.5, V = 0.5, W = -0.5, U = -0.5, C = 1)
  ),
  outcome = "Y",
  exposure = "X",
  latent = "C"
) %>% 
  # Convert the DAG into a tibble
  tidy_dagitty() %>% 
  # Set node status
  node_status()

Closing Backdoors

Next we can use the adjustmentSets from the dagitty package to get the nodes we need to adjust for to identify the path \(\treat{X}_{\tim{t}} \longrightarrow \resp{Y}_{\tim{t}}\)

# Get the minimum adjustment set
quota_dag$dag %>% 
  adjustmentSets()
{ V, W, Z }

Closing Backdoors

Next we can use the adjustmentSets from the dagitty package to get the nodes we need to adjust for to identify the path \(\treat{X}_{\tim{t}} \longrightarrow \resp{Y}_{\tim{t}}\)

# Get the minimum adjustment set
quota_dag$dag %>% 
  adjustmentSets()
{ V, W, Z }

Then we use adjust_for from ggdag to adjust for the variables in the output and use mutate to wrangle some new columns for color and transparency

# Adjust for the variables in the output
quota_dag_adjusted <- quota_dag %>% 
  adjust_for(var = c("V", "W", "Z")) %>% 
  # Create a columns color coding the nodes and edges
  mutate(status_color = case_when(
    is.na(status) | status == "latent" ~ as.character(adjusted),
    TRUE ~ as.character(status)
    )) %>% 
  # Adjust the length of the arrows slightly
  shorten_dag_arrows(., proportion = 0.08)

Closing Backdoors

# Generate the DAG for the contemporaneous effect of X on Y
ggplot(quota_dag_adjusted, aes(x = x, y = y, xend = xend, yend = yend)) +
  # Add the graph edges
  geom_dag_edges(
    aes(x = xstart, y = ystart, edge_alpha = if_else(status_color == "adjusted", 0, 1)), 
    edge_width = 1.5,
    edge_color = "white",
    arrow_directed = grid::arrow(length = grid::unit(10, "pt"), type = "closed"),
    show.legend = FALSE
  ) +
  # Add the graph nodes
  geom_dag_node(alpha = 0) +
  # Add the graph text
  geom_dag_text(
    aes(label = label, color = status_color),
    parse = TRUE,
    size = 22,
    family = "serif",
    show.legend = FALSE
  ) +
  # Apply the theme settings for the slides
  dag_theme(.base_size = 24) +
  # Set the color scale for the text
  scale_color_manual(values = c("#B21A01", "#00F2FF", "#62FC6B", "#9C9C9C"))

Closing Backdoors

Intermediate Outcomes

  • \(\treat{X}\) causes \(\resp{Y}\)

  • But \(\treat{X}\) also causes \(\covar{Z}\) which has a causal effect on \(\resp{Y}\)

  • Should you adjust for \(\covar{Z}\)?

  • No! In this case \(\covar{Z}\) is a mediator or an intermediate outcome

  • Adjusting for an intermediate outcome induces post-treatment bias (Montgomery, Nyhan, and Torres 2018)

Colliders and Selection Bias

  • \(\treat{X}\) causes \(\covar{Z}\)

  • \(\resp{Y}\) also causes \(\covar{Z}\)

  • Should you adjust for \(\covar{Z}\)?

  • No! In this case \(\covar{Z}\) is a collider, a common effect of both \(\treat{X}\) and \(\resp{Y}\)

  • Colliders can create a fake causal effects or hide real ones

  • Conditioning on a collider induces endogenous selection bias

Reproducing Barnes and Holman (2020)

The Journal of Politics (2020)
Gender Quotas, Women's Representation, and Legislative Diversity
Tiffany D. Barnes; Mirya R. Holman

How Quotas Influence Legislative Diversity

  • The adoption of gender quotas contributes to the erosion of gendered understandings of “candidate quality” and leads to the expansion of party recruitment networks

  • Over time, this leads leads to both increases in female representation–the goal of gender quotas–but also serves to increase legislative diversity by changing candidate recruitment

  • Authors theorize about a causal mechanism, a complex process by which things influence one another both directly and indirectly to produce some outcome

  • They test their argument on subnational data from Argentina between 2004 and 2018

Replication in R

  • Download the replication data for the article from the JOP dataverse archive

    • If you installed git when you were installing R and RStudio, you can just type git pull in the terminal window in RStudio from within the course’s R project to get the data and R code

    • If you haven’t installed git, you should install git because it will allow you to automatically sync your local files with updates I make to the course repository instead of having to download them from Canvas (i.e., for problem sets, data, code, and lecture slides)

Packages and Settings

# Set Session Options
options(
  digits = 6, # Significant figures output
  scipen = 999, # Disable scientific notation
  repos = getOption("repos")["CRAN"],
  knitr.kable.NA = '',
  brms.backend = "cmdstanr",
  modelsummary_get = "broom"
)

Packages and Settings

# Set Session Options
options(
  digits = 6, # Significant figures output
  scipen = 999, # Disable scientific notation
  repos = getOption("repos")["CRAN"], # Install packages from CRAN
  knitr.kable.NA = '', # Set NA values to blank when making tables
  brms.backend = "cmdstanr", # Use cmdstanr as a backend for {brms}
  modelsummary_get = "broom" # Custom tiody method for modelsummary
)

# Load the necessary libraries
pacman::p_load(
  "sjlabelled", # Package for working with labeled data
  "tidyverse", # Suite of packages for data management 
  "haven", # Reading in data from proprietary software (Stata, SPSS, SAS) 
  "brms", # Bayesian regression models with Stan
  "tidybayes", # Functions for wrangling posteriors tidy-style
  "modelsummary", # Package for making tables of model output
  "kableExtra", # Package for customizing html and latex tables
  "patchwork", # Combining multiple plots into one
  "future", # Package for parallel computation
  install = FALSE
)

Helper Functions

Assuming you’re in the course’s project directory, you should be able to load the custom functions we’ll use here with the following code

# Source the helper functions, assumes you're working in the course's R project
.helpers <- map(
  .x = list.files(
    path = "functions/", 
    pattern = ".*R", 
    full.names = TRUE
  ), 
  .f = ~ source(.x)
)

Reading and Processing Data

We need to use the read_dta function from haven since the replication data is in Stata’s proprietary file format

# Load the replication data
jop_data <- read_dta("data/Barnes_and_Holman_JOP2020.dta")

Reading and Processing Data

We need to use the read_dta function from haven since the replication data is in Stata’s proprietary file format

# Load the replication data
jop_data <- read_dta("data/Barnes_and_Holman_JOP2020.dta")

# Extract the data for the models
model_data <- jop_data %>% 
  # Select will return a subset with just the specified variables
  select(chamber_year:senate, diversity_person:logdm) %>% 
  # Get the time index
  mutate(
    # Apply value labels from chamber year
    chamber_year = factor(
      chamber_year,
      levels = get_values(chamber_year),
      labels = get_labels(chamber_year)
    ),
    # Get the year from chamber_year
    year = str_extract(chamber_year, "20[0-9][0-9]") %>% as.integer()
  )

Specifying the Models

  • Barnes and Holman (2020) consider six different versions of their outcome, legislative diversity. For each outcome \(\resp{y}\), their model specification can be expressed as

\[ \begin{align*} \resp{y}_{\obs{i}} &\sim \mathcal{N(\mean{\mu}_{\obs{i}}, \sigma^{2})}\\ \mean{\mu}_{\obs{i}} =& \alpha + \beta_{1} \mathrm{Female~Representation} + \beta_{2}\mathrm{Time~Since~Quota} + \\ & \quad \beta_{3} \mathrm{Gender-Related~Development} + \beta_{4}\mathrm{Unemployment} + \\ & \quad \beta_{5}\mathrm{Log~District~Magnitude} + \beta_{6}\mathrm{Senate~Chamber} + \sigma\\ \end{align*} \]

Specifying the Models

We need to start by building the model formulas and to save time we’ll use a custom function formula_builder that we loaded earlier

# Character vector of response variables for each model in table 1 on pp. 1279
responses <- c(
  "prof_diversity", 
  "diversity_person", 
  "prof_diversity_f", 
  "diversity_person_f",
  "prof_diversity_m",
  "diversity_person_m"
)

Specifying the Models

We need to start by building the model formulas and to save time we’ll use a custom function formula_builder that we loaded earlier

# Character vector of response variables for each model in table 1 on pp. 1279
responses <- c(
  "prof_diversity", 
  "diversity_person", 
  "prof_diversity_f", 
  "diversity_person_f",
  "prof_diversity_m",
  "diversity_person_m"
)

# Formula for the LHS which are the same across models in table 1
rhs_form <- "~female + quotayear + logdm + senate + unemployment + gdi"

Specifying the Models

We need to start by building the model formulas and to save time we’ll use a custom function formula_builder that we loaded earlier

# Character vector of response variables for each model in table 1 on pp. 1279
responses <- c(
  "prof_diversity", 
  "diversity_person", 
  "prof_diversity_f", 
  "diversity_person_f",
  "prof_diversity_m",
  "diversity_person_m"
)

# Formula for the LHS which are the same across models in table 1
rhs_form <- "~female + quotayear + logdm + senate + unemployment + gdi"

# Build the list of formulas
model_forms <- formula_builder(lhs = responses, rhs = rhs_form)

Priors for the Model

We need to specify priors on the model parameters (more on this in later classes) and its usually a good idea to save each model to local storage so you don’t have to run things again

# Specify the priors object to pass to brm
model_priors <- prior(normal(0, 1.5), class = "b") +
  prior(normal(mean(Y), 2*sd(Y)), class = "Intercept") +
  prior(exponential(1/sd(Y)), class = "sigma")

# Build a list of file paths to save the models to
model_paths <- str_c(
  "models/Barnes-and-Holman-2020/gaussian_", 
  responses, 
  "_full"
)

Fitting the Models

Since I’m lazy, we’re just going to fit each model iteratively using the map function from purrr. Basically, for each index in the list object model_forms, we fit the model and the result is a list of model objects of the same length

# Fit each of the models (6 chains, 5k iterations)
bayes_gaussian_fits <- map(
  .x = seq_along(model_forms),
  .f = ~ brm(
    formula = bf(model_forms[[.x]], decomp = "QR"),
    family = brmsfamily("gaussian", link = "identity"),
    prior = model_priors,
    data = model_data,
    cores = 6, # Adjust chains and cores based on your computer's hardware
    chains = 6, # Should be at least 4
    iter = 5000,
    warmup = 3000,
    refresh = 1000,
    control = list(max_treedepth = 12),
    save_pars = save_pars(all = TRUE),
    seed = 12345,
    backend = "cmdstanr",
    file = model_paths[.x]
  )
)

Fitting the Models

The looking code is equivalent to that on the previous slide but has the disadvantage of being much less elegant

# Initialize a list ot store the models in
bayes_gaussian_fits <- list()

# Fit each of the models (6 chains, 5k iterations)
for (i in seq_along(model_forms)) {
  bayes_gaussian_fits[[i]] <- brm(
    formula = bf(model_forms[[i]], decomp = "QR"),
    family = brmsfamily("gaussian", link = "identity"),
    prior = model_priors,
    data = model_data,
    cores = 6,
    chains = 6,
    iter = 5000,
    warmup = 3000,
    refresh = 1000,
    control = list(max_treedepth = 12),
    save_pars = save_pars(all = TRUE),
    seed = 12345,
    backend = "cmdstanr",
    file = model_paths[i]
  )
}

References

Alexander, Amy C. 2012. “Change in Women’s Descriptive Representation and the Belief in Women’s Ability to Govern: A Virtuous Cycle.” Politics & Gender 8: 437–64.
Barnes, Tiffany D., and Mirya R. Holman. 2020. “Gender Quotas, Women’s Representation, and Legislative Diversity.” The Journal of Politics 82(4): 1271–86.
Esarey, Justin, and Leslie A. Schwindt-Bayer. 2019. “Estimating Causal Relationships Between Women’s Representation in Government and Corruption.” Comparative Political Studies 52: 1713–41.
Greenland, Sander, Judea Pearl, and James M. Robins. 1999. “Causal Diagrams for Epidemiologic Research.” Epidemiology 10(1): 37–48.
Hughes, Melanie M., Mona Lena Krook, and Pamela Paxton. 2015. “Transnational Womens Activism and the Global Diffusion of Gender Quotas.” International Studies Quarterly 59(2): 357–72.
Irvine, Jill. 2007. “From Civil Society to Civil Servants: Women’s Organizations and Critical Elections in Croatia.” Politics & Gender 3(1).
Montgomery, Jacob M., Brendan Nyhan, and Michelle Torres. 2018. “How Conditioning on Posttreatment Variables Can Ruin Your Experiment and What to Do about It.” American Journal of Political Science 62(3): 760–75.
Paxton, Pamela, and Melanie M. Hughes. 2015. “The Increasing Effectiveness of National Gender Quotas, 1990-2010.” Legislative Studies Quarterly 40(3): 331–62.
Paxton, Pamela, Melanie M. Hughes, and Jennifer L. Green. 2006. “The International Womens Movement and Womens Political Representation, 18932003.” American Sociological Review 71(6): 898–920.
Paxton, Pamela, Melanie M. Hughes, and Matthew A. Painter. 2010. “Growth in Women’s Political Representation: A Longitudinal Exploration of Democracy, Electoral System and Gender Quotas.” European Journal of Political Research 49: 25–52.
Schwindt-Bayer, Leslie. 2009. “Making Quotas Work: The Effect of Gender Quota Laws on the Election of Women.” Legislative Studies Quarterly 34(1): 5–28.
Stockemer, Daniel. 2011. “Women’s Parliamentary Representation in Africa: The Impact of Democracy and Corruption on the Number of Female Deputies in National Parliaments.” Political Studies 59: 693–712.
Tripp, Aili Mari, and Alice Kang. 2008. “The Global Impact of Quotas: On the Fast Track to Increased Female Legislative Representation.” Comparative Political Studies 41: 338–61.
VanderWeele, Tyler J., and James M. Robins. 2007. “Directed Acyclic Graphs, Sufficient Causes, and the Properties of Conditioning on a Common Effect.” American Journal of Epidemiology 166(9): 1096–1104.